The Shapley Value for Stochastic Cooperative Game

The research is financed by the foundation for the edbiz of He Bei province of China(2004468)and the foundation for the natural science of He Bei province of China(A2005000301) Abstract In this paper we extend the notion of Shapley value to the stochastic cooperative games. We give the definition of marginal vector to the stochastic cooperative games and we define the Shapley value for this game. Furthermore, we discuss the axioms of the Shapley value and give the proofs of these axioms.


Introduction
In general, the payoffs of a coalition in cooperative games are assumed to be known with certainty.In many cases, however, payoffs to coalitions are uncertain.If the formation of coalitions and allocations has to take place before the payoffs are realized, standard cooperative game theory can not be applied.Suijs et al. (1995) considered cooperative games with stochastic payoffs, the model introduced by Suijs et al. (1995) is explicitly incorporates preferences on stochastic payoffs for each agent and allows each coalition to choose from several actions.Suijs et al. (1999) continue on the model introduced by Suijs et al. (1995).They extend the definitions of superadditivity and convexity for TU games to stochastic cooperative games.Furthermore, they show that a stochastic cooperative game has a nonempty core.
In this paper we take the model introduced by Suijs et al. (1999) as a basis.We define the Shapley value of stochastic cooperative games.Furthermore, we discuss the axioms of the Shaley value.This paper is organized as follows.In section2 we introduce basic definitions concerning stochastic cooperative games.Section3 presents our main results.The axioms for Shapley value of stochastic cooperative game.

Stochastic cooperative games
Let us first recall some of the definitions concerning stochastic cooperative games as introduced by Suijs et al. (1999).A stochastic cooperative game is described by a tuple where N is the set of agents, S A the nonempty and finite set of actions a coalition S can take, L R of stochastic payoffs with finite expectation.We assume that for each player the preferences are complete, transitive and continuous.Furthermore, we assume that ( ( ) 0) 1 P X a for all a A .The class of all cooperative games with stochastic payoffs with agent set N is denoted by ( ) SG N .To simplify notation, however, we restrict our attention to the case that each coalition only has one action to take, that is, for all S N .So we can denote a stochastic cooperative game by ( ,{ } ,{ } ) For our definition of the Shapley value of stochastic cooperative game, we first give the definition of marginal vector.

Let ( , )
N v be a game and let N be the set of all permutations of N .Then the kth coordinate of the marginal vector Now we extend the notion of marginal vector to the stochastic cooperative games. Let be a stochastic cooperative game, and let N be the set of all permutations of N. Then the kth coordinate of the marginal vector Before we give the definition of Shspley value of stochastic cooperative game, we define some useful notions.
1. (Carrier) Let be a stochastic cooperative game, T N is called a carrier for the game if There have two properties of the carrier: (1).Let T be the carrier of , then for all T is the carrier of too.
Because, for all S N , we have (2).Let T be the carrier of , i T , then for all S N , we have We can describe the Shapley value as the average of the marginal vectors of the player i to the coalitions.
Then we extend the Shapley value to the stochastic cooperative games.
N X is a stochastic cooperative game, then the Shapley value for the player i ( (1)

Axioms for the Shapley value
We give one axiomatization for the Shapley value of stochastic cooperative game.We consider the axioms: for all i N .

(Additivity) If and
( ) SG N , we have ( ) ( ) ( ) Then the Shapley value of stochastic cooperative game is the vector that satisfies the axioms above.Proof.(1) Efficiency axiom.Let T be the carrier of , according to the property (2) of the carrier, we have (2).Symmetry axiom.Let be a permutation of N , which satisfies formula (2), then we have S S , and for While \ i N T , T and T i are carriers of T cX , then according to efficiency axiom, we have ( ) , for all i T .
While , i j T , and i j , let be a permutation of N , such that , , , , , , .
First we prove  for all i N .And we know the number T c is unique derived by T and S X , hence ( ) i is unique derived by S X , N and i , that is to say the Shapley value ( ) is unique derived by S X and N .
payoff function of coalition S, assigning to each action R with finite expectation, and i the preference relation of agent i over the set 1 ( ) player) The player k N is a dummy in the stochastic cooperative game ( ,{ } ,{ } ) is easy to prove that T cX is the characteristic function, and T is a carrier of T cX .